From (1) and (3), it follows that e0 has a cause e1, which in turn has a cause e2, and so on, in an infinite regress.

From (2) we know that there can be no infinite regress, which contradicts (4).

Therefore, at least one of the premises must be false.

If we reject premise 1, that every event has a cause, then there must be at least one uncaused cause, which can be called "God".

Why call it God?

Even if we accept the argument from first cause, the conclusion is still problematic: the word "God" carries a lot of undesirable cultural baggage, denoting an intelligent being. If the ultimate cause of our universe turns out to be, say, a random vacuum fluctuation, then that would be "God" by Aquinas's definition, but to call this phenomenon "God" would be misleading. It also can be noted that if for some reason there did have to be a first cause, we don't know what it was or enough about it to give it attributes other than being the first cause, which would make calling it and assuming it is God an argument from incredulity or the "God of the gaps" idea.

Argument from incredulity

To say that because we currently observe cause and effect relationships occuring in space-time, the universe itself must have had a first cause is assuming something that no one knows. Before the big bang time could not be said to exist at least in the way that we perceive it, and so to make assumptions about the behavior of matter prior to time is pure speculation. Simply because one cannot conceive of something happening without a cause, does not mean that we can assume everything needs a cause. The logic fails as soon as we attempt to preceed the planck time, after which nothing can accurately be said about the relationship of cause and effect. It is much the same as saying that order and design must mean there was a designer, just because as far as you can tell this is the case with things that humans design. You are saying you cannot fathom a natural process that would explain what you are trying to explain and so you assume that nature must abide by your narrow understanding of order and design. One's narrow understanding of cause and effect based on their current concept of time does not mean that before time was as it is today their narrow views still apply.